Variations of Yoneda Lemma; Monos, Epis and Isomorphisms of (Pre)sheaves

The first part is my work on a variation of Yoneda Lemma. The second part is my work on Exercises 2.4A, 2.4.C-2.4.D of section 2.4 in Vakil’s notes.

1.  Variations of Yoneda Lemmas (Monos, Epis and Isomorphisms of Presheaves)

Here are a few variations of Yoneda Lemma I played around a few years ago, which bear similar ideas of Yoneda Lemma. Recently I have been dealing with sheaves again, so I just reviewed some old stuff here. I used these variations to show that a morphism of presheaves is monic resp. epic if and only if it’s injective resp. surjective on the level of sections (For another proof see here https://stacks.math.columbia.edu/tag/00V5).

Here are the variations and proofs for presheaves: (pdf version: Variations of Yoneda lemma)

2.  Monos, Epis and Isomorphisms of Sheaves

Here we give a detailed discussion for sheaves, following exercises in Section 2.4 of Vakil’s notes: (pdf version: Monos, epis and isomorphisms of sheaves)

Categorical descriptions for glueing sheaves and schemes

In this post, we give a categorical proof of Lemma 33.2 (Tag 00AK) by showing that a glued sheaf is defined as an equaliser. This equaliser provides a tool for calculating global sections of glued schemes. We later present some examples for calculating global sections and glueing constructions using (co)limits descriptions. (Sometimes people say they may lose insights about the details and the real maths behind abstraction. But it really depends on one’s approach and ways of thinking. ) The goal of this post is to give a structured summary of glueing constructions of schemes after meditating on explicit constructions, using categorical language. It is not meant to replace explicit arguments for schemes, but to give some ideas on how general a construction is and whether it can be transferred to a different setting. For example, we will know what to do if we are working on a site with a different Grothendieck topology instead of the Zariski topology. Note that some consequences of the categorical facts we use are indeed straightforward for schemes, for example, the two pullback squares in Example 4. This post is open-ended and more examples of glueing will be added.

Remark. Note that for a sheaf $\mathcal {F}$ on a topological space and an open cover $U=\bigcup U_i,$

$\displaystyle \mathcal {F}(U) =\mathrm{lim}_J \mathcal{F}(U_i),$

where $J$ is the covering sieve generated by the covering $\{U_i\}$, namely, $J$ is the collection of all those $V\subset U$ with $V\subset U_i$ for some $i.$ This limit is equivalent to the following equaliser diagram in the usual sheaf definition:

where for $t\in FU$, $e(t)=\{t|_{U_i} \mid i\in I\}$ and for a family $t_i\in FU_i, p\{t_i\}=\{t_i| _{U_i\cap U_j}\}, q\{t_i\}=\{t_j| _{U_i\cap U_j}\}.$

For a scheme $X=\bigcup X_i$ with an open cover $\{X_i\}$ in the Zariski topology, $X$ is the colimit indexed over the covering sieve generated by the covering $\{X_i\}$. This colimit can also be simplified to be a coequaliser diagram.

First, we include the explicit constructions for glueing sheaves and some sources for details check of these constructions. The categorical proof we are going to show is a categorical rephrasing by meditating on these constructions, which gives a more structured presentation.

Glueing Morphisms

Proposition A.1  [Tag 00AK] Let $X$ be a topological space. Let $X=\cup U_i$ be an open covering. Let $\mathcal{F},\mathcal{G}$ be sheaves of sets on $X$. Given a collection

$\phi_i:\mathcal{F}|_{U_i}\to \mathcal {G}|_{U_i}$

of maps of sheaves such that for all $i,j\in I$ the maps $\phi_i,\phi_j$ restrict to the same map $\mathcal {F}_{U_i\cap U_j}\to \mathcal{G}_{U_i\cap U_j}$, then there exists a unique map of sheaves

$\phi: \mathcal {F}\to \mathcal{G}$

whose restriction to  each $U_i$ agrees with $\phi_i$.

Proof. Take any $s\in \mathcal{F}(V)$, where $V\subset X$ is open, and let $V_i=U_i\cap V$. Then we have an element $\phi_i (s| _{V_i})\in \mathcal{F}(V_i)$ and $\phi_i (s| _{V_{ij}})=\phi_j (s| _{V_{ij}})$ by the glueing condition. Thus by the sheaf condition for $G$, the sections $\phi_i(s|_{V_i})\in G(V_i)$ patch together to give a section in $G(V)$, define this section to be $\phi(s)$. (We omitted the checking details. )                                                   $\square$

Glueing Sheaves

Explicit construction of glueing sheaves is given in Lemma 6.33.2, Tag 00AK, but the details of checking have been omitted.  For some details of a reality check,  see this post.

Glueing Schemes

To glue schemes, one needs to define the glued topological spaces which will be a quotient space of the disjoint union of the glued spaces and then verify the structure sheaves of the glued schemes satisfy the condition of Lemma 6.33.2 [see Tag 01JA].

Example 1. (The affine line with doubled origin is not affine).   Let $k$ be a field. Let $X= \text{Spec} (k[t])$, $Y= \text{Spec}(k[u])$. Let $U=D(t) =\text{Spec}k[t,1/t]\subset X \text{ and } V= D(u)=\text{Spec}k[u,1/u]\subset Y$.  Consider the ismorphism $U\cong V$ given by $t\leftrightarrow u$.  Let $Z$ be the glued scheme, from the equaliser diagram in the proof of Lemma 6.33.2,  we see that the structure sheaf $\mathcal{O}_Z$ is given by

$\mathcal{O}_{Z}(W) = \mathcal{O}_X(W \cap X) \times_{\mathcal{O}_{X(W \cap X \cap Y)} \cong \mathcal{O}_{Y}(W \cap X \cap Y)} \mathcal{O}_Y(W \cap Y)$.

Thus the global section $\mathcal{O}_Z(Z) \text{ is } k[t] \times_{k[t,t^{-1}] \cong k[u,u^{-1}]} k[u] \cong k[t]$. From this we see that $Z$ is not affine, since  $Z=\text{Spec}(k[t])$ which is not the case: the underlying topological space $Z$ has one more point- the doubled origin.

Example 2. (Quasiseparated scheme is glued from affine schemes). Note that every scheme is a colimit of affine schemes. This is true in general by the fact that every sheaf is a colimit of representables and that the Zariski topology is subcanonical (see here for more details).

For the case that a scheme $X$ is separated (for which the intersection
of any two affine open sets is affine), take an affine open cover $\bigcup U_i=X$ such that each intersection $U_{ij}$ is affine, then $X$ is just the coequaliser of the diagram $\coprod_{i,j} U_{ij} \rightrightarrows \coprod_{i} U_i$. This is the glueing construction as we described above.

If $X$ is not separated, one can still write $X$ as a colimit of affines but not with the same diagram as we used for glueing, see here for a description of the diagram.

Remark. One implication of viewing $X$ as a colimit is: let Sch and Rings be the categories of schemes and rings respectively, given that $\text{Hom}_{\textbf{Rings}}(A,B)\cong \text{Hom}_{\textbf{Sch}}(\text{spec}(B),\text{spec}(A))$, one can deduce that for any scheme $X$$\text{Hom}_{\textbf{Rings}}(R,\Gamma(X,\mathcal{O}_x))\cong \text{Hom}_{\textbf{Sch}}(X,\text{spec}(R))$ by the fact Hom-functor preserves (co)limits.

Let $S_\bullet=\oplus _{n\in \mathbb{Z}} S_n$ be a $\mathbb{Z}$-graded ring and $S_+=\oplus_{i>0} S_i$ be the irrevalant ideal. Suppose $f\in S_+$  is homogeneous, there is a bijection between the prime ideals of $((S_\bullet)_f)_0$ and the homogeneous prime ideal of $(S_\bullet)_f$. The projective distinguished open set $D(f)= \mathrm{Proj} S_\bullet \setminus V(f)$ is identified with $\mathrm{Spec}((S_\bullet)_{f})_0$. If $f,g\in S_{+}$ are homogeneous and nonzero, $D(f)\cap D(g)= \mathrm{Spec} ((S_\bullet)_{fg})_0)$ is isomorphic to the distinguished open subset $D(g^{\mathrm{deg} f}/f^{\mathrm{deg}g})$ of $\mathrm{Spec} ((S_\bullet)_f)_0$, similarly for $\mathrm{Spec} ((S_\bullet)_g)_0$. $\mathrm{Proj}S_\bullet$ is glued from various $\mathrm{Spec}((S_\bullet)_{f})_0$ along the pairwise intersections $\mathrm{Spec}((S_\bullet)_{fg})_0$.
Given arbitrary schemes $X,Y,S$, let $q:X\to S$ and $r: Y\to S$ be the given morphisms. Let $\{S_i\}$ be an open affine cover of $S$. Let $X_i=q^{-1}(S_i)$, $Y_i=r^{-1}(S_i)$, choose an affine open cover $X_{ij}$ for $X_i$ and an affine open cover $Y_{jk}$ for $Y_k$. The fibre product is constructed by glueing various $X_i\times_{S_i} Y_i$  together.